Recently, graph convolutional network (GCN), as a powerful graph embedding technology, has been widely applied in the field of recommendation. The main reason is that most of the information in recommender systems can be modeled as graph-structured data, and GCN, as a deep learning model that operates on graph structures, helps to explore the potential interactions between users and items in graph-structured data, to enhance the performance of the recommender systems. Since the modeling of recommender systems usually needs to collect and process a large amount of sensitive data, it may face the risk of privacy leakage. Differential privacy, as a privacy protection model with a solid theoretical foundation, has been widely used in recommender systems to solve the problem of personal privacy leakage. Currently, the research based on differential privacy is mainly oriented to independent and identically distributed data models. However, data within GCN-based recommender systems is highly correlated and not independent, making the existing privacy protection methods less effective. To solve the problem, this study proposes a graph convolutional collaborative filtering recommendation algorithm based on Rényi differential privacy (RDP-GCF for short), aiming to achieve a balance between privacy protection and utility while ensuring the security ofuser-item interaction data. The algorithm first utilizes GCN techniques to learn the embedding vectors for users and items. Then, the Gaussian mechanism is used to randomize the embedding vectors, and a sampling-based method is used to amplify the privacy budget and minimize the injection of differential noise, thereby improving the performance of the recommender system. Lastly, the final embedding vectors of the users and items are obtained by a weighted fusion and applied to the recommendation tasks. The proposed algorithm is validated through experiments on three publicly available datasets. The results show that compared to existing similar methods, the proposed algorithm more effectively achieves a balance between privacy protection and data utility.